Problem: $ A = \left[\begin{array}{r}4 \\ 1 \\ 4\end{array}\right]$ $ C = \left[\begin{array}{rr}2 & -2 \\ 2 & 2 \\ 2 & 2\end{array}\right]$ Is $ A- C$ defined?
In order for subtraction of two matrices to be defined, the matrices must have the same dimensions. If $ A$ is of dimension $( m \times  n)$ and $ C$ is of dimension $( p \times  q)$ , then for their difference to be defined: 1. $ m$ (number of rows in $ A$ ) must equal $ p$ (number of rows in $ C$ ) and 2. $ n$ (number of columns in $ A$ ) must equal $ q$ (number of columns in $ C$ Do $ A$ and $ C$ have the same number of rows? Yes Yes No Yes Do $ A$ and $ C$ have the same number of columns? No Yes No No Since $ A$ has different dimensions $(3\times1)$ from $ C$ $(3\times2)$, $ A- C$ is not defined.